Intro to probability and statistics.(if you solve you a genius)

Question

Question 2: We have a collection of steel circles. The radius of each circle is a realization of a random variable R uniformly distributed in [1, 5].



1. a) Determine the expected value of the volume of the circles. Note: A= π R2 (10 pts)
2. b) Is it true that E(A) = π [E(R)]2 ? Explain your answer. (8 pts)

Answer 1

For a given R in [1,5] consider the interval [R, R +dR] where dR is small

Probability the radius is in R, R+dR is dR/[5-1] = dR/4

Corresponding area is πR²

So to find the expected area, multiply probability by area and sum from R=1 to 5 (so integrate)

So 1/4 πR² integrated from R=1 to R=5 so π/4 [ R³/3 ] = π/12 [ 5³ - 1³]

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E(R) = (5-1) / 2 = 3

So clearly this isn't true.